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ON PROXIMATE ORDER AND PROXIMATE TYPE OF ENTIRE DIRICHLET SERIES
ARKOJYOTI BISWAS
Abstract. In this paper we introduce the notion of Proximate Order and Proximate Type of Entire Dirichlet Series and prove their existence. We also obtain some related results.
1. Introduction
Let f (s) be an entire function of the complex variable s = + it de…ned by the everywhere convergent Dirichlet series
f (s) = 1 X n=1 anes n (1) where 0 < n < n+1(n 1); n! 1 as n ! 1 and an2 C:
If cand abe respectively the abscissa of convergence and absolute convergence
of (1) then c= a = 1:
For an entire function f (s) represented by (1) its maximum modulus is denoted by F ( ) and is de…ned as
F ( ) = sup fjf ( + it)j : t 2 Rg : The Ritt order f of f (s) is de…ned as
f = lim sup !1
log[2]F ( ) where ( cf. [5]):
log[k]x = log(log[k 1]x) for k = 1; 2; 3 ... and log[0]x = x: Received by the editors: Oct. 17, 2015, Accepted: Jan. 04, 2016.
2010 Mathematics Subject Classi…cation. Primary: 30B50, Secondary:30D99.
Key words and phrases. Entire Dirichlet series, order, type, proximate order and proximate type.
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For …nite Ritt order f the type Tf of f (s) is de…ned as
Tf = lim sup !1
log F ( )
e f :
During past decades several authors {see [1],[2],[3],[4]} made close investigations on various properties of the entire Dirichlet series. Therefore with a view to obtain sharper estimation of the growth properties of f (s) when f is …nite,we …rst intro-duce the concept of the proximate order and then prove its existence in the line of Shah [6].Let us …rst de…ne the proximate order of an entire function represented by Dirichlet series.
De…nition 1. Let f (s) be an entire function represented by Dirichlet series (1) with …nite order f.A function ( ) is said to be a proximate order of f if ( ) has the following properties:
: (i) ( ) is non-negative and continuous for > 0, say,
: (ii) ( ) is di¤erentiable for 0 except possibly at isolated points at
which 0( + 0) and 0( 0) exist, : (iii) lim !1 ( ) = f, : (iv) lim !1 0( ) = 0 and : (v) lim sup !1 log F ( ) exp f ( )g = 1.
Since the type Tf is not linked with the proximate order we may expect another
comparison function which should closely connect the type and the maximum mod-ulus of an entire function represented by Dirichlet series (1).With this in view we de…ne and prove the existence of such a function in line of Srivastava and Juneja [7] which we call proximate type of f (s):
We now de…ne the proximate type of an entire function represented by Dirichlet series.
De…nition 2. For an entire function f (s) represented by (1) with …nite order f and …nite type Tf, a function T ( ) is said to be a proximate type of f if T ( ) has
the following properties:
: (i) T ( ) is non-negative and continuous for > 0, say,
: (ii) T ( )is di¤erentiable for 0 except possibly at isolated points at
which T0( + 0) and T0( 0) exist,
: (iii) lim !1T ( ) = Tf, : (iv) lim !1 T 0( ) = 0 and : (v) lim sup !1 F ( ) expfT ( )e fg = 1. 2. Theorems
Theorem 1. Let f (s) be an entire function represented by Dirichlet series (1) with …nite Ritt order f .Then the proximate order ( ) of f (s) exists.
Proof. Let p ( ) = log[2]F ( ):Then lim sup
!1
p ( ) = f: We consider two cases:
Case(I) : Let p ( ) > f for at least a sequences of values of r tending to in…nity. we de…ne
( ) = max
x fp (x)g:
Clearly ( ) exists and is non increasing.
Let R > eeand p (R) > f. Then for R1> R say, we get p ( ) p (R) :Since
p ( ) is continuous, there exists 12 [R; R1] such that
p ( 1) = max
R x R1fp (x)g:
Clearly 1 > ee and ( 1) = p ( 1) :Such values = 1 exists for a sequence of
values of tending to in…nity.
Let ( 1) = ( 1) and t1be the smallest integer not less than 1 + 1 such that
( 1) > (t1):
We de…ne ( ) = ( 1) for 1 < t1 .Observing that ( ) and ( 1)
log[2] + log[2]t1 are continuous functions of , ( 1) log[2] + log[2]t1 > (t1)
for (> t1) su¢ ciently close to t1 and ( ) is non increasing,we can de…ne u1 as
follows: u1> t1;
( ) = ( 1) log[2] + log[2]t1 for t1 u1;
( ) = ( ) for = u1and
( ) > ( ) for t1 < u1:
Let 2 be the smallest value of for which 2 u1 and ( 2) = p ( 2) :If 2 > u1 then let ( ) = ( ) for u1 2:Since it can be easily shown that
( ) is constant in u1 2; ( ) is constant in u1 2:We repeat this
process in…nitely and obtain that ( ) is di¤erentiable in adjacent intervals. Further
0( ) = 0 or ( log ) 1
and ( ) ( ) p ( ) for all 1:Also ( ) = p ( )
for a sequences of values of tending to in…nity and ( ) is non increasing for
1 and f = lim sup !1 p ( ) = lim !1 ( ) : So lim sup !1 ( ) = lim inf
!1 ( ) = lim!1 ( ) = f and lim!1
Further we have log[2]F ( ) = p ( ) = ( ) for a sequence of values of tending to in…nity and log[2]F ( ) < ( ) for remaining ’s. Therefore
lim sup
!1
log F ( ) exp f ( )g = 1:
Continuity of ( ) for 1 follows from its construction which is complete in
case(I).
Case(II) : Let p ( ) f for all su¢ ciently large values of r. In Case(II) we separate two cases:
Sub case (A) : Let p ( ) = f for at least a sequence of values of tending to in…nity:
Sub case (B) : Let p ( ) < f for all su¢ ciently large values of . In Sub case (A) we take ( ) = f for all su¢ ciently large values of . In Sub case (B) let
( ) = max
X x fp (x)g;
where X > ee is such that p ( ) <
f whenever x X. We note that ( ) is non
decreasing and for all X su¢ ciently large, the roots of (x) = f + log[2]x
log[2] is less than . For a suitable large value v1> X; we de…ne
(v1) = f;
( ) = f+ log[2] log[2]v1for s1 v1where s1< v1is such that (s1) =
(s1).In fact s1is given by the largest positive root of (x) = f+log [2]
x log[2]v1.
If (s1) = p (s1) ; let 1(< s1) be the upper bound of point at which ( ) 6= p ( )
and < s1:Clearly at 1; (s1) = p (s1) :We de…ne ( ) = ( ) for 1 s1.
It is easy to show that ( ) is constant in 1 s1 and so ( ) is constant in 1 s1. If (s1) = p (s1) we take 1= s1:
We choose v2> v1 suitably large and let
(v1) = ;
( ) = f + log[2] log[2]v2 for s2 v2 where s2 < v2 is such that
(s2) = (s2) :If (s2) 6= (s2) let ( ) = ( ) for 2 s2;where 2 has the
similar property as that of 1:As above ( ) is constant in[ 2; s2]. If (s2) = p (s2)
we take 2= s2:
Let ( ) = ( 2) log[2] + log[2] 2 for q1 2 where q1 < 2 is the
point of intersection of y = f with y = ( 2) log[2]x + log[2] 2:It is also possible
to choose v2 so large that v1 < q1:Let ( ) = f for v1 q1:We repeat this
process. Now we can show that for all v1; f ( ) ( ) p ( ) and
( ) = p ( ) for = 1; 2; ::: .So we obtain that
lim sup
!1
( ) = lim inf
Since log[2]F ( ) = p ( ) = ( ) for a sequence of values of tending to in…nity and log[2]F ( ) < ( ) for remaining ’s it follows that
lim sup
!1
log F ( ) exp f ( )g = 1:
Also ( ) is di¤erentiable in adjacent intervals. Further 0( ) = 0 or ( log ) 1
and so
lim
!1
0( ) = 0:
Continuity of ( ) follows from its construction. This completes the proof of the theorem.
Corollary 1. exp f ( )g is an increasing function for > 0:
Theorem 2. Let f (s) be an entire function represented by Dirichlet series (1) with …nite Ritt order f and …nite type Tf .Then the proximate type T ( ) of f (s) exists.
Proof. Let s ( ) = log F ( ) e f : Then lim sup !1 s ( ) = Tf:
Then either Case (A): Let s ( ) > Tf for at least a sequences of values of tending
to in…nity or,
Case (B):Let s ( ) Tf for all su¢ ciently large values of .
In Case (A) we de…ne
( ) = max
x fs (x)g:
Clearly ( ) exists and is non increasing. Let R > eeand s (R) > T
f . Then for R1> R say, we get s ( ) s (R) :Since
s ( ) is continuous, there exists 12 [R; R1] such that
s ( 1) = max
R x R1fs (x)g:
Clearly 1 > ee and ( 1) = s ( 1) :Such values = 1 exists for a sequence of
values of tending to in…nity.
Let T ( 1) = ( 1) and p1be the smallest integer not less than 1 + 1such that
( 1) > (p1):
We de…ne T ( ) = T ( 1) for 1 < p1 .Observing that ( ) and T ( 1)
log[2] + log[2]p1 are continuous functions of , T ( 1) log[2] + log[2]p1> (p1)
for (> p1) su¢ ciently close to p1 and ( ) is non increasing,we can de…ne u1 as
follows: u1> p1;
T ( ) = T ( 1) log[2] + log[2]p1 for p1 u1;
T ( ) > ( ) for p1 < u1:
Let 2 be the smallest value of for which 2 u1 and ( 2) = s ( 2) :If 2 > u1 then let T ( ) = ( ) for u1 2:Since it can be easily shown that
( ) is constant in u1 2; T ( ) is constant in u1 2:We repeat
this process in…nitely and obtain that T ( ) is di¤erentiable in adjacent intervals. Further T0( ) = 0 or ( log ) 1 and T ( ) ( ) s ( ) for all 1:Also
T ( ) = s ( ) for a sequences of values of tending to in…nity and T ( ) is non increasing for 1and
Tf = lim sup !1 s ( ) = lim !1 ( ) : So lim sup !1 T ( ) = lim inf
!1 T ( ) = lim!1T ( ) = Tf and lim!1 T
0( ) = 0:
Further we have F ( ) = exp fs ( ) e f g = exp fT ( ) e f g for a sequence of values
of tending to in…nity and F ( ) < exp fT ( ) e f g for remaining ’s;.Therefore
lim sup
!1
F ( )
expfT ( )e fg = 1:
Continuity of T ( ) for 1 follows from its construction which is complete in
Case(A).
In Case(B) we separate two cases:
Sub case (I):Let s ( ) = f for at least a sequence of values of tending to
in…nity:
Sub case (II):Let s ( ) < f for all su¢ ciently large values of .
In Sub case (I) we take T ( ) = f for all su¢ ciently large values of .
In Sub case (II) let
( ) = max
X x fs (x)g;
where X > ee is such that s ( ) <
f whenever x X. We note that ( ) is non
decreasing and for all X su¢ ciently large, the roots of (x) = f + log[2]x log[2] is less than . For a suitable large value v1> X; we de…ne
T (v1) = f;
T ( ) = f+ log[2] log[2]v1for s1 v1where s1< v1is such that (s1) =
T (s1).In fact s1is given by the largest positive root of (x) = f+log[2]x log[2]v1.
If (s1) = T (s1) let !1(< s1) be the upper bound of point ! at which (!) 6= s (!)
and ! < s1:Clearly at !1; (s1) = s (s1) :We de…ne T ( ) = ( ) for !1 s1.
It is easy to show that ( ) is constant in !1 s1and so T ( ) is constant in
!1 s1. If (s1) = s (s1) we take !1= s1:
We choose v2> v1 suitably large and let
T (v1) = f;
T ( ) = f + log[2] log[2]v2 for s2 v2 where s2 < v2 is such that
similar property as that of !1:As above T ( ) is constant in[!2; s2]. If (s2) = T (s2)
we take !2= s2:
Let T ( ) = T (!2) log[2] + log[2]!2 for q1 !2 where q1 < !2 is the
point of intersection of y = T with y = T (!2) log[2]x + log[2]!2:It is also possible
to choose v2 so large that v1 < q1:Let T ( ) = f for v1 q1:We repeat this
process. Now we can show that for all v1; f T ( ) ( ) s ( ) and
T ( ) = s ( ) for = !1; !2; ::: .So we get that
lim sup
!1
T ( ) = lim inf
!1 T ( ) = lim!1T ( ) = f:
SinceF ( ) = exp fs ( ) e f g = exp fT ( ) e f g for a sequence of values of
tend-ing to in…nity and F ( ) < exp fT ( ) e f g for remaining ’s;we get that
lim sup
!1
F ( )
expfT ( )e fg = 1:
Also T ( ) is di¤erentiable in adjacent intervals. Further T0( ) = 0 or ( log ) 1 and so
lim
!1 T
0( ) = 0:
Continuity of T ( ) follows from its construction. This completes the proof of the theorem.
Theorem 3. Let T ( ) be the proximate type of f (s):Then lim inf
!1
log T ( ) = 0: Proof. As lim sup
!1
F ( )
expfT ( )e fg = 1; for arbitrary " > 0 and for sequence of values
of we get
(1 ") expfT ( )e fg F ( )
i:e:; log T ( ) + f+ O (1) log[2]F ( ) i:e:;log T ( )+ f+O (1) log
[2]
F ( )
i:e:; lim inf
!1 log T ( ) + f lim sup !1 log[2]F ( ) = f
i:e:; lim inf
!1
log T ( ) 0: Since T ( ) is non negative it follows that
lim inf
!1
log T ( ) = 0: This completes the proof of the theorem.
References
[1] Q.I.Rahaman : The Ritt order of the derivative of an entire function, Anales Polonici Mathe-matici,17(1965), 137-140.
[2] C.T.Rajagopal and A.R.Reddy : A note on entire functions represented by Dirichlet series, Aanales Polonici Mathematici,17(1965), 199-208.
[3] J.F.Ritt : On certain points in the theory of Derichlet series, Amer. Jour. Math., 50(1928), 73-86.
[4] R.P Srivastav and R.K.Ghosh : On entire functions represented by Dirichlet series, Aanales Polonici Mathematici,13(1963), 93-100.
[5] D. Sato : On the rate of growth of entire functions of fast growth, Bull. Amer. Math. Soc., 69 (1963), 411-414.
[6] S.M. Shah : On proximate orders of integral functions, Bull. Amer. Math. Soc.,Vol.52 (1946), pp.326-328.
[7] R.S.L. Srivastava and O.P. Juneja : On proximate type of entire functions,Composito Mathematica,18(1967),7-12.
Current address : Ranaghat Yusuf Institution P.O.-Ranaghat, Dist-Nadia, PIN-741201,West Bengal, India.
